The number of values of k, for which the system of equations `(k""+""1)x""+""8y""=""4k` `k x""+""(k""+""3)y""=""3k-1` has no solution, is (1) 1 (2) 2 (3) 3 (4) infinite

Options

A. infinite
B. 1
C. 2
D. 3

Answer

D

Explanation

Given equations can be written in matrix form AX =B
where, `A = [{:(k+1, 8), (k, k+3):}], X = [(x)/(y)] " and "B = (4k)/(3k-1)`
For no solution, |A|=0 and (adj A) `B ne 0`
Now, `|A| = [{:(k+1, 8), (k, k+3):}] =0`
`rArr (k^(2) + 1)(k+3)-8k =0`
`k^(2) +4k+3-8k=0`
`rArr k^(2)-4k xx 3 =0`
`rArr(k-1)(k-3) =0`
`rArr k =1, k =3`,
Now, adj `A = [{:(k+3, -8), (-k, k+1):}]`
Now (adj A)`B = [{:(k+3, -8), (-k, k+1):}][{:(4k), (3k+1):}]`
` = [{:((k+3)(4k) -8(3k-1)), (-4k^(2)+ (k+1)(3k-1)):}]`
`=[{:(4k^(2) -12k+8), (-k^(2)+ 2k-1):}]`
Put k = 1
(adj A)`B=[{:(4-12+8), (-1+2-1):}] = [(0), (0)]` not true
Put k =3
`("adj A") B=[{:(36-36+8), (-9+6-1):}] = [(8), (-4)] ne 0` true
Hence, required value of k is 3.
Alternate Solution
Condition for the system of equations has no solution is
`(a_(1))/(a_(2)) = (b_(1))/(b_(2)) ne (c_(1))/(c_(2))`
`therefore (k+1)/(k) = (8)/(k+3) ne (4k)/(3k-1)`
Take `(k+1)/(k) = (8)/(k+3)`
`rArr k^(2) + 4k + 3 =8k`
`rArr k^(2) - 4k + 3`
`rArr (k-1)(k-3) = 0`
`k =1, 3`
If `k-1, "then" (8)/(1+3) ne (4.1)/(2)`, false
And, if `k = 3, "then" (8)/(6) ne (4.3)/(9-1)`, true
Therefore, k = 3
Hence, only one value of k exist.